Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / O / 2 / i / Solution

With immediate replacement and uniform mixing, each capture is a Bernoulli trial with tagged probability $p=k/N$. Counting failures before the $r$th tagged capture gives the stated negative binomial distribution. Independent runs on the successive days give the sample model.

Writing $S={\sum }_i{x}_i$, the log-likelihood is $\ell (p)=nr\log p+S\log (1-p)+\text{constant}$. Its derivative vanishes atThe second derivative is negative. If $S=0$, this formula gives the boundary maximum $p=1$; on the strict physical parameter space $p<1$, it is a supremum rather than an attained interior maximum.

Using the supplied negative-binomial mean, $\mathbb{E}S=nr(1-p)/p$. Thus total Fisher information for the sample is